Aren't scores in the Wasserstein GAN probabilities?
Figure 3 in the original WGAN paper is actually quite helpful to understand the difference between the score in WGAN and the probability in GAN (see screenshot below). The blue distribution are real samples, and the green one are fake samples. The Vanilla GAN trained in this example identifies the real samples as '100% real' (red curve) and the fake samples as '100% fake'. This leads to the problem of vanishing gradients and the well-known mode collapse of original GANs.
The Wasserstein GAN, on the other hand, gives each sample a score. The benefit of the score is that we can now identify samples that are more likely real than others, or more likely fake. For example, the further to the left a distribution is located, the more negative the WGAN score will be. We have therefore a continuum that doesn't end in 0 and 1, but can compare between samples that are 'good' and those which are 'better'. A normal GAN would identify both as 'good', making further improvement difficult.